Show that if the diagonals of a quadrilateral are equal and bisect each other at right angles, then it is a square

Explanation

From`/_\AOB and/_\COD`
`AO = CO`(Diagonals bisect each other)
`/_AOB = /_COD`(Vertically opposite)
`OB=OD`(Diagonals bisect each other)
`/_\AOB ~=/_\ COD`(SAS congruency)
Thus,`AB = CD`(CPCT) (i)
`/_OAB=/_OCD`(Alternate interior angles)
`=> AB || CD`
In `/_\AOD and /_\COD`
`AO=CO`(Diagonals bisect each other)
`/_AOD=/_COD`(Vertically opposite)
`OD=OD`(Common)
`/_\AOD~=/_\COD` (SAS congruency)
`AD = CD`(CPCT)(ii)
`AD=BC\and\AD=CD`
`=>AD=BC=CD=AB`(ii)
also,`/_ADC=/_BCD`(CPCT)
`/_ADC+/_BCD 180^@` (co-interior angles)
`=> 2/_ADC = 180^@`
`=>/_ADC=90^@`(iii)
One of the interior angles is right angle.
Thus, from (i), (ii) and (iii) given quadrilateral ABCD is a square.